ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Pages 90–94 (October 5, 2000) S 1079-6762(00)00085-8

A ONE-BOX-SHIFT MORPHISM BETWEEN SPECHT MODULES

MATTHIAS KÜNZER

(Communicated by David J. Benson)

Abstract. We give a formula for a morphism between Specht modules over (Z/m)S_{n}, where n ≥ 1, and where the partition indexing the target Specht module arises from that indexing the source Specht module by a downwards shift of one box, m being the box shift length. Our morphism can be reinterpreted integrally as an extension of order m of the corresponding Specht lattices.

0. Notation

We write composition of maps on the right, ^{\overline{}^{α}} ^{\overline{}^{β}} = ^{\overline{}^{αβ}} . Intervals are to be read as subsets of Z. Let n ≥ 1, let S_{n} = Aut_{Sets}[1, n] denote the symmetric group on n letters and let ε_{σ} denote the sign of a permutation σ ∈ S_{n}. Let

[figure: N; λ; N_{0}; i; λ_{i}]

be a partition of n, i.e. assume ∑_{i} λ_{i} = n and λ_{i} ≥ λ_{i+1} for i ∈ N. Let

[λ] := {i × j ∈ N × N | j ≤ λ_{i}}

denote the diagram of λ. We say that i × j ∈ [λ] lies in row i and in column j. A λ-tableau is a bijection

[figure: [λ]; [a]; ∼; [1, n]; i × j; a_{i,j}.]

The element σ ∈ S_{n} acts on the set T^{λ} of λ-tableaux via composition [a] ^{\overline{}^{σ}} [a]σ. Let F^{λ} be the free Z-module on T^{λ} with the induced operation of S_{n}. Let

[figure: [λ]; ρ; N; [λ]; κ; N; i × j; i; i × j; j]

denote the projections. We denote by {a} := [a]^{−1}ρ the λ-tabloid associated to the λ-tableaux [a]. The free Z-module on the set of tabloids, equipped with the inherited S_{n}-operation, is denoted by M^{λ}. Let

C_{[a]} := {σ ∈ S_{n} | [a]^{−1}κ = ([a]σ)^{−1}κ}
\begin{matrix}c \\ ◯ \end{matrix}2000 American Mathematical Society

Received by the editors July 14, 2000.

2000 Mathematics Subject Classification. Primary 20C30.

Key words and phrases. Symmetric group, Specht module.

be the column stabilizer of [a]. Let the Specht lattice S^{λ} be the ZS_{n}-sublattice of M^{λ} generated over Z by the λ-polytabloids

⟨a⟩ := ∑_{σ∈C_{[a]}}{a}σε_{σ}.

Let λ′ denote the transposed partition of λ, i.e. j ≤ λ_{i} ⟺ i × j ∈ [λ] ⟺ i ≤ λ′_{j}.

1. Carter-Payne

Let d ∈ [1, n] be the number of shifted boxes. Let 1 ≤ s < t ≤ n, s being the row of [λ] from which the boxes are shifted, and t being the row into which the boxes are shifted. Suppose

μ_{i} := \{ \begin{matrix}λ_{i} − d & for i = s, \\ λ_{i} + d & for i = t, \\ λ_{i} & else \end{matrix}

defines a partition of n. Let the box shift length be denoted by

m := (λ_{s} − s) − (λ_{t} − t) − d.

Let m[p] := p^{v_{p}(m)} be the p-part of m. Using [1], Carter and Payne proved the following

Theorem 1.1 ([2]). Let K be an infinite field of characteristic p. Suppose d < m[p]. Then

Hom_{KS_{n}}(K ⊗_{Z} S^{λ}, K ⊗_{Z} S^{μ}) ≠ 0.

2. Integral reinterpretation

Assume d = 1, i.e. [μ] arises from [λ] by a one-box-shift. The condition d < m[p] translates into p|m.

As we will see below, this particular case of the result of Carter and Payne already holds over K = F_{p}. So we obtain a nonzero element in

Hom_{ZS_{n}}(S^{λ}/pS^{λ}, S^{μ}/pS^{μ}) ^{\overline{}^{∼}} Hom_{ZS_{n}}(S^{λ}, S^{μ}/pS^{μ}).

We consider a part of the long exact Ext_{ZS_{n}}^{∗}(S^{λ}, −)-sequence on

0 \overline{}S^{μ} ^{\overline{}^{p}} S^{μ} \overline{}S^{μ}/pS^{μ} \overline{}0,

viz.

0 \overline{}\underbrace{Hom_{ZS_{n}}(S^{λ}, S^{μ})}_{= 0} ^{\overline{}^{p}} \underbrace{Hom_{ZS_{n}}(S^{λ}, S^{μ})}_{= 0} \overline{}Hom_{ZS_{n}}(S^{λ}, S^{μ}/pS^{μ})

\overline{}Ext_{ZS_{n}}^{1}(S^{λ}, S^{μ}) ^{\overline{}^{p}} Ext_{ZS_{n}}^{1}(S^{λ}, S^{μ}).

Mapping our morphism into Ext^{1}, we obtain a nonzero element of Ext_{ZS_{n}}^{1}(S^{λ}, S^{μ}) which is annihilated by p. Conversely, the p-torsion elements of Ext^{1} are given by morphisms modulo p.

Since n! annihilates Ext_{ZS_{n}}^{1}(S^{λ}, S^{μ}), replacement of p by n! shows that any element in Ext^{1} is given by a modular morphism modulo n!,

Hom_{ZS_{n}}(S^{λ}, S^{μ}/n! S^{μ}) ^{\overline{}^{∼}} Ext_{ZS_{n}}^{1}(S^{λ}, S^{μ}).

Therefore, in order to get hold of the whole Ext^{1}, we need to calculate modulo prime powers in general.

3. One-box-shift formula

We keep the assumption d = 1. Let s′ := λ_{s} and let t′ := λ_{t} + 1. A path of length l ∈ [1, s′ − t′] is a map

[figure: [0, l]; γ; [λ] ∪ [μ]; k; α_{k} × β_{k}]

such that k < k′ implies β_{k} < β_{k′}, and such that α_{0} × β_{0} = t × t′ and β_{l} = s′. For a λ-tableau [a], we define the μ-tableau [a^{γ}] by

a_{i,j}^{γ} := a_{i,j}  for i × j ∈ [μ]\(γ([1, l]) ∪ N × {s′}),
a_{α_{k},β_{k}}^{γ} := a_{α_{k+1},β_{k+1}} for k ∈ [0, l − 1],
a_{i,s′}^{γ} := a_{i,s′}  for i < α_{l},
a_{i,s′}^{γ} := a_{i+1,s′}  for i ≥ α_{l}.

For i ∈ [t′ + 1, s′ − 1], we denote

X_{i} := (s′ − λ′_{s′}) − (i − λ′_{i}).

Let

x_{γ} := (−1)^{α_{l}+1} \frac{∏_{i∈[t′+1,s′−1], μ′_{i}>μ′_{i+1}} X_{i}}{∏_{k∈[1,l−1]} X_{β_{k}}} .

Let Γ be the set of paths of some length l ∈ [1, s′ − t′].

Theorem 3.1 ([4], 4.3.31, cf. 0.7.1). The abelian group Hom_{ZS_{n}}(S^{λ}, S^{μ}/mS^{μ}) contains an element f of order m = (λ_{s} − s) − (λ_{t} − t) − 1 which is given by the commutative diagram of ZS_{n}-linear maps

[figure: [a]; ∑_{γ∈Γ} x_{γ} ⊗ ⟨a^{γ}⟩; F^{λ}; Q ⊗_{Z} S^{μ}; [a]; ⟨a⟩; S^{μ}; ⟨b⟩; ⟨b⟩ + mS^{μ}.; 1 ⊗ ⟨b⟩; S^{λ}; f; S^{μ}/mS^{μ}]

Reducing modulo a prime dividing m, this recovers the case d = 1 of the result of Carter and Payne. By the long exact sequence as above, but with p replaced by m, we obtain a nonzero element in Ext_{ZS_{n}}^{1}(S^{λ}, S^{μ}) of order m. ^{1}

The proof of this theorem proceeds by showing that a sufficient set of Garnir relations in F^{λ} is annihilated by F^{λ} \overline{}S^{μ}/mS^{μ}.

1I do not know the structure of Ext_{ZS_{n}}^{1}(S^{λ}, S^{μ}) as an abelian group. At least in case n ≤ 7, direct computation yields that the projection of our element to its 2′-part generates this 2′-part.

We have, however, for example Ext_{ZS_{6}}^{1}(S^{(4,1^{2})}, S^{(3,1^{3})})_{(2)} ≃ Z/2 ⊕ Z/2.

4. Example

Let n = 9, λ = (4, 3, 2), μ = (3, 3, 2, 1), t′ = 1 and s′ = 4, whence m = 6, X_{2} = 4, X_{3} = 2. We obtain a morphism of order 6 that maps

S^{(4,3,2)}  ^{\overline{}^{f}} S^{(3,3,2,1)}/6 S^{(3,3,2,1)}

⟨ \begin{matrix}1 & 4 & 7 & 9 \\ 2 & 5 & 8 \\ 3 & 6 \end{matrix} ⟩ \overline{}4^{0}2^{0} (⟨ 1 7 9 2 5 8 3 6 4 ⟩ + ⟨ 1 4 9 2 7 8 3 6 5 ⟩ + ⟨ 1 4 9 2 5 8 3 7 6 ⟩

+⟨ 1 8 7 2 5 9 3 6 4 ⟩ + ⟨ 1 4 7 2 8 9 3 6 5 ⟩ + ⟨ 1 4 7 2 5 9 3 8 6 ⟩)

+4^{1}2^{0} (⟨ 1 4 9 2 5 8 3 6 7 ⟩ + ⟨ 1 4 7 2 5 9 3 6 8 ⟩)

+4^{0}2^{1} (⟨ 1 9 7 2 5 8 3 6 4⟩ + ⟨ 1 4 7 2 9 8 3 6 5⟩ + ⟨ 1 4 7 2 5 8 3 9 6⟩)

+4^{1}2^{1} (⟨ 1 4 7 2 5 8 3 6 9⟩) .

The [0, l − 1]-part of the respective path is highlighted.

5. Motivation

We consider the rational Wedderburn isomorphism

[figure: QS_{n}; ∼; ∏_{λ}(Q)_{n_{λ}×n_{λ}}; σ; (ρ_{σ}^{λ})_{λ}]

where λ runs over the partitions of n and where ρ_{σ}^{λ} denotes the matrix describing the operation of σ ∈ S_{n} on S^{λ} with respect to a chosen tuple of integral bases. The restriction

ZS_{n} [figure] ∏_{λ}(Z)_{n_{λ}×n_{λ}}

of this isomorphism, viewed as an embedding of abelian groups, has index ^{2}

∏_{λ} (\frac{n!}{n_{λ}} )^{n_{λ}^{2}/2} .

2Question. Given a central primitive idempotent e^{λ} of Γ := ∏_{λ}(Z)_{n_{λ}×n_{λ}}, what is the index of e^{λ}ZS_{n} in e^{λ}Γ ? Cf. ([4], Section 1.1.3).

In particular, for n ≥ 2 it is no longer an isomorphism.

Suppose, for partitions λ and μ of n and for some modulus m ≥ 2, we are given a ZS_{n}-linear map

S^{λ} ^{\overline{}^{g}} S^{μ}/mS^{μ}.

Let G be the matrix, with respect to the chosen integral bases of S^{λ} and S^{μ}, of a lifting of g to a Z-linear map S^{λ} \overline{}S^{μ}. The ZS_{n}-linearity of g reads

Gρ_{σ}^{μ} − ρ_{σ}^{λ}G ∈ m(Z)_{n_{λ}×n_{μ}} for all σ ∈ S_{n}.

Thus such a morphism yields a necessary condition for a tuple of matrices to lie in the image of the Wedderburn embedding.

For example, the evaluations of our one-box-shift morphism at hook partitions, i.e. at λ = (k, 1^{n−k}) and μ = (k − 1, 1^{n−k+1}), k ∈ [2, n], furnish a long exact sequence. In the (simple) case of n = p prime, and localized at (p), the set of necessary conditions imposed by these morphisms already turns out to be sufficient for a tuple of matrices over Z_{(p)} to lie in the image of the localized Wedderburn embedding ([4], Section 4.2.1). Therefore, it is advisable to chose a tuple of locally integral bases adapted to this long exact sequence. For instance, we obtain

\begin{aligned}Z_{(3)}S_{3} & ^{\overline{}^{∼}} & \{a × [ \begin{matrix}b & c \\ d & e \end{matrix} ] × f | a ≡_{3} b, d ≡_{3} 0, e ≡_{3} f\} \\ & ⊆ & Z_{(3)} × [ \begin{matrix}Z_{(3)} & Z_{(3)} \\ Z_{(3)} & Z_{(3)} \end{matrix} ] × Z_{(3)}, \end{aligned}

the embedding not being written in the combinatorial standard polytabloid bases.

For an approach to the general case, see ([4], Chapters 3 and 5). Further examples may be found in ([4], Chapter 2).

6. Acknowledgments

This result being part of my thesis, I’d like to repeat my thanks to my advisor Steffen König.

References

1. R. W. Carter and G. Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 139–242. MR 50:7364

2. R. W. Carter and M. T. J. Payne, On homomorphisms between Weyl modules and Specht modules, Math. Proc. Camb. Phil. Soc. 87 (1980), 419–425. MR 81h:20048

3. G. D. James, The representation theory of the symmetric groups, SLN 682, 1978. MR 80g:20019

4. M. Künzer, Ties for the ZS_{n}, thesis, http://www.mathematik.uni-bielefeld.de/∼kuenzer, Bielefeld, 1999.

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld

   E-mail address: kuenzer@mathematik.uni-bielefeld.de